PROPOSITION XXXV. PROBLEM.

583. To find the volume of a spherical segment.

Let A B D C generate a spherical segment by revolving about A C.

To find the volume of the segment.

SUG. 1. Join B and D to the center of the semicircle O.

SUG. 2. Find the volume of the spherical sector generated by revolving the circular sector B O D. (Ex. 379.) SUG. 3. Add the volume of the cone generated by revolving the ABAO.

SUG. 4. Subtract the volume generated by revolving the ADCO.

Ex. 883. Find the volume of a spherical sector having a zone of 10 in. altitude on a sphere of 20 in. radius.

Ex. 384. Find the volume of a spherical segment of 8 in. altitude, one base of which passes through the center of the sphere, the radius of the sphere being 20 in.

EXERCISES.

385. A sphere is cut by parallel planes so that the diameter is divided into ten equal parts: compare the area of the zones; also the volumes of the spherical sectors whose spherical surfaces are the respective zones.

386. If the diameter of the sphere is 30 ft., find the volumes of the first, fifth, and seventh segments referred to in exercise 385.

387. Find the volume and area of the surface of the sphere in exercise 386.

388. The sides of a triangle, on a sphere whose radius is 10 ft., are, respectively, 95°, 117°, and 37°. Find the area in square feet of its polar triangle.

389. Assuming the earth to be a sphere whose diameter is 7912 miles, how many square miles upon its surface.

390. If the average specific gravity of the earth is 74, what is its weight expressed in tons.

391. Assuming the diameter of the earth to be 8,000 miles, and that of the moon 2,000; how do the amounts of light reflected from them to a point in space equally distant from each compare?

392. With the same assumption as that of exercise 391, what is the ratio of the volumes of the earth and moon?

393. A triangle on a 12 in. globe has for its angles 140°, 119°, and 196° respectively; compute its area.

394. The diameter of a sphere is equal to the altitude of a cone of revolution and of a cylinder of revolution, and the radius of the sphere is equal to the radius of the cone and of the cylinder. Prove that the volumes of the cone, sphere, and cylinder are proportional to the num bers 1, 2, and 3.

PROPOSITIONS IN CHAPTER VIII.

PROPOSITION I.

Every section of a cylinder made by a plane containing an element is a parallelogram.

PROPOSITION II.

The bases of a cylinder are equal.

PROPOSITION III.

The area of the lateral surface of a cylinder is equal to the perimeter of a right section multiplied by an element of the surface.

PROPOSITION IV.

The volume of a cylinder equals the area of a right section multiplied by a lateral edge.

PROPOSITION V.

Every section of a cone made by a plane passing through the ver tex is a triangle.

PROPOSITION VI.

Every section of a circular cone made by a plane parallel to the base is a circle.

PROPOSITION VII.

The area of the convex surface of a cone of revolution is equal to one half the product of the perimeter of its base by its slant height.

PROPOSITION VIII.

The area of the convex surface of a frustum of a cone of revolution is equal to one half the product of its slant height by the sum of the perimeters of its bases.

PROPOSITION IX.

The volume of a cone equals one third of the product of the area of its base by its altitude.

PROPOSITION X.

Every section of a sphere made by a plane is a circle.

PROPOSITION XI.

Two intersecting great circles of a sphere bisect each other.

PROPOSITION XII.

Three points on the surface of a sphere determine a circle of the sphere.

PROPOSITION XIII.

The shortest distance on the surface of a sphere between any two points on that surface is the arc, not greater than a semi-circumference, of a great circle which joins them.

PROPOSITION XIV.

All points in the circumference of a circle are equally distant from either of its poles.

PROPOSITION XV.

Given a material sphere, to find its radius.

PROPOSITION XVI.

A plane perpendicular to a radius of a sphere at its extremity, is tangent to the sphere.

PROPOSITION XVII.

A spherical angle is equal to the dihedral angle formed by the planes of the two arcs, and is measured by the arc of a great circle described from the intersection of the arcs as a pole, and intercepted between them.

PROPOSITION XVIII.

Two symmetrical isosceles spherical triangles can be made to coincide, and are equal.

PROPOSITION XIX.

If a spherical triangle D E F is the polar of another spherical triangle A B C, then the triangle A B C is the polar of D E F.

PROPOSITION XX.

In two polar triangles each angle of one is measured by the sup plement of the side opposite it in the other.

PROPOSITION XXI.

Two triangles, on the same sphere, or equal spheres, having twc sides and the included angle of one equal respectively to two sides and the included angle of the other, are either equal or symmetrical.

PROPOSITION XXII.

Two triangles, on the same sphere, or equal spheres, having two angles and the included side of one equal respectively to two angles and the included side of the other, are either equal or symmetrical

PROPOSITION XXIII.

Two triangles, on the same sphere or equal spheres, having the hree sides of one respectively equal to the three sides of the other, are either equal or symmetrical.

PROPOSITION XXIV.

Two triangles on the same sphere, or equal spheres, having the three angles of one respectively equal to the three angles of the other are either equal or symmetrical.

PROPOSITION XXV.

The sum of the sides of a convex spherical polygon is less than the circumference of a great circle

PROPOSITION XXVI.

The sum of the angles of a spherical triangle is greater than two and less than six right angles.

PROPOSITION XXVII.

Two symmetrical spherical triangles are equal in area.

PROPOSITION XXVIII.

If two arcs of great circles intersect on the surface of a hemisphere, the sum of the areas of the opposite spherical triangles thus formed is equal to the area of a lune whose angle is the angle of the intersecting arcs.